Exploring the self-mode locking and vortex structures of nonplanar elliptical modes in selectively end-pumped Nd:YVO4 lasers: manifestation of large fractional orbital angular momentum

An end-pumped Nd:YVO4 laser under selective pumping is used to excite lasing modes with transverse patterns performed to exhibit the characteristics of multiple spots arranged on elliptical features near degenerate cavities. The spatial distribution of elliptical lasing modes is clearly revealed to be localized on the nonplanar ray orbits, so-called nonplanar elliptical modes, which possess large fractional orbital angular momentum. Moreover, temporal dynamics for the output emission of nonplanar elliptical modes are verified to obtain self-mode-locked operation. We further numerically manifest not only the influence of radial-asymmetry distributions on the vortex structures of nonplanar elliptical modes, but also the vector field of transverse lasing modes altered with twisting phase structures in the propagation direction. © 2017 Optical Society of America OCIS codes: (140.4050) Mode-locked lasers; (050.4865) Optical vortices; (140.3480) Lasers, diode-pumped. References and links 1. 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Introduction
Techniques of self-mode locking with a decrease of cavity length for developing highrepetition-rate optical pulses is an interesting phenomenon without using active elements or saturable absorbers [1]. The efficient self-mode-locked operation has recently been realized in various diode-pumped Yb-doped [2][3][4][5] and Nd-doped [6-9] crystal lasers. In particular, the self-mode-locked high-order Hermite-Gaussian (HG) modes were experimentally confirmed in a selectively diode-end-pumping scheme [10]. Moreover, it has been found that when the cavity length is adjusted to be the degenerate cavities, the self-mode locking with the simultaneously longitudinal and transverse mode locking can be achieved. To fulfill the degenerate cavities, the cavity configuration requires satisfying the degenerate condition of ∆f T /∆f L = P/Q, where P and Q are co-prime integers, ∆f T is the longitudinal mode spacing, and ∆f L is the transverse mode spacing. The simultaneously self-mode locking can be seen to be a time-dependent wave packet with dynamical behaviors corresponding to the planar periodic ray trajectories inside the cavity, known as planar geometric modes [11,12]. Thus far the optical configuration producing these multipass beams has been widely used in diverse experiments, such as optical delay lines [13], absorption spectroscopy [14][15][16], Raman conversion [17][18][19][20], high-power laser systems [21] and low repetition rate ultrafast laser cavities [22][23][24].
However, the generalized multipass beams in a three-dimensional (3D) laser resonator near the degenerate cavities can be localized on the nonplanar ray orbits, described as nonplanar geometric modes. The transverse pattern of nonplanar geometric modes is performed to demonstrate the property of multiple spots arranged on the topological trajectories. The topological trajectories are altered with the relative phase φ between the degenerate eigenmodes from the line feature at 0 φ = (i.e., planar geometric modes) through the elliptical shape in the range of 0 /2 φ π < < and to the circular form at / 2 φ π = . Hence, the nonplanar geometric modes can be generally represented by nonplanar elliptical modes. More interestingly, these nonplanar elliptical modes exhibit the existence of light fields with spiral characteristics during propagation. Several properties of spiral optical beams developed by Abramochkin and Volostnikov [25,26] are of practical interest for laser technologies, medicine, and microbiology. Specifically, in recent years elliptical modes with possessing fractional orbital angular momentum (OAM) and lacking for radial symmetry have been observed to be a rapidly growing interest in the field of quantum communication to achieve high security of quantum cryptography and quantum teleportation [27][28][29][30][31][32][33][34]. It is believed that the investigation of nonplanar elliptical modes with the self-mode-locked operation and the spiral features carrying huge fractional OAM is an important step in the structure light to be exploited in numerous applications.
In this work we exploit a diode-end-pumped Nd:YVO 4 laser with selective pumping to produce the nonplanar elliptical modes near the degenerate cavities. The experimental farfield patterns of nonplanar elliptical modes with the characteristics of multiple spots reveal to obtain the self-mode locking. To analyze the average OAM and the vortex structures of nonplanar elliptical modes, we theoretically perform that spatial distributions of nonplanar elliptical modes can be perfectly reconstructed with the numerical calculations. Theoretical exploration validates the influence of radial-asymmetry distributions on the vortex structures of nonplanar elliptical modes that the primary singularities can be separated by controlling the phase factor between the degenerate eigenmodes. It is further observed that the vector field of transverse lasing modes is displayed swirling features along the elliptical structure at the beam waist, and evidently altered with the twisting phase angle field in the propagation direction.

Experimental results for the self-mode locking of nonplanar elliptical modes
Here we demonstrate nonplanar elliptical modes generated from nearly degenerate cavities by exploiting the selective pumping in a concave-plano resonator as shown in Fig. 1. The front mirror is a 10-mm radius-of-curvature concave mirror with antireflection coating at 808 nm on the entrance face and with high-reflectance coating at 1064 nm (>99.8%) and high transmittance coating (T>95%) at 808 nm on the second surface. The gain medium is an a-cut 2.0-at.% Nd 3+ :YVO 4 crystal with a length of 2 mm. One side of the gain medium is coated for antireflection at 808nm and 1064 nm (reflection < 0.1%) and the other side is coated to be an output coupler with a transmission of 0.5% at 1064 nm. The pump source is an 808-nm fibercoupled laser diode with a core diameter of 100 μm, a numerical aperture of 0.16, and a maximum output power of 3 W. A focusing lens with 25 mm focal length and 85% coupling efficiency is used to focus the pump beam into the gain medium. The pump radius is estimated to be approximately 25 μm. We use the lens to reimage the transverse patterns at the different longitudinal positions inside the cavity. . Since the cavity mode possessing the biggest overlap with the gain region will dominate the laser emission, the nonplanar elliptical mode can be generated by precisely manipulating and persistently shifting the pump position on the gain medium. The experimental results of the nonplanar elliptical mode for (P,Q) = (1,3) can be found under the off-center pumping with accurately transverse displacements of ∆x = 0.47 mm and ∆y = 0.45 mm along the x-and y-directions; the nonplanar elliptical mode for (P,Q) = (1,5) is produced with ∆x = 0.43 mm and ∆y = 0.80 mm along the x-and y-directions, respectively. On the whole, nonplanar elliptical modes are usually difficult to be flexibly generated in the present approach that the strictly transverse displacements are required to balance the astigmatism induced by the birefringence of the gain medium and to achieve the isotropic effective cavity lengths in x-and y-directions.  Figure 4 shows the schematic of the time-dependent photon wave packet related to the mode-locked operation with dynamical behaviors corresponding to the nonplanar elliptical modes in the laser cavity. It indicates that when the ratio ∆f T /∆f L is a simple fraction P/Q with P and Q to be co-prime integers, the wave packet inside the cavity periodically returns to its original position and direction after ( ) P Q ⋅ round trips for the round-trip time t r = 2L/c. It has recently been confirmed that when the cavity length is adjusted to be the degenerate cavity, the self-mode locking can be achieved to generate a localized wave packet traveling along the planar geometric trajectories inside the cavity by identifying the relative shifts of the time sequences between different spots of the far-field pattern [11,12]. Here a similar approach is utilized to build a self-mode-locked laser in Fig. 1 demonstrating the signal to noise to be greater than 35 dB. The fundamental frequency peaks can be seen to be 4.529 GHz in Fig. 5(c) and 4.289 GHz in Fig. 5(d), which correspond accurately to ∆f T shown that a wave packet returns exactly to the same position and direction after 3 and 5 round trips in the cavity, respectively. Since nonplanar elliptical modes require simultaneously longitudinal and transverse mode locking, the mode-locked condition is certainly more rigorous than fundamental modes with purely longitudinal mode locking. Even though the performance of mode locking for nonplanar elliptical modes is not perfect as the fundamental modes, it can be seen that the pulse trains display full modulation without CW background, indicating that the mode-locked operation is achieved. It is worthwhile to mention that the above results are all for the isotropic effective cavity lengths in x-and ydirections inside the laser resonator. To increase the effective path length exploited in numerous applications, the present approach can be further justified to introduce an appropriate astigmatism into the laser cavity to achieve mode-locked states corresponding to nonplanar periodic ray orbits with marvelous structures.

Theoretical analysis of vortex structures generated from nonplanar elliptical modes
Considering the paraxial approximation, the eigenmodes for the laser cavity with a concave mirror at z = -L and a plane mirror at z = 0 can be divided into two waves traveling in opposite directions: and 1/ 2  Fig. 3 with ∆x = 0.43 mm and ∆y = 0.80 mm. The parameter φ is provided with / 5 π due to the ellipticity of the lasing modes in Fig. 2 and Fig. 3. Therefore, the average OAM can be calculated as 139.3  and 255.1  for (P,Q) = (1,3) in Fig. 2 and (P,Q) = (1,5) in Fig. 3, respectively, which indicate the large fractional OAM in such beams. However, the integer N related to the experimental observations to be 237 for (P,Q) = (1,3) and 462 for (P,Q) = (1,5) is so high that it is difficult to be analyzed by numerical calculations. Figure 6 just reveals the numerical transverse patterns to illustrate the contours of the experimental patterns. The upper row in Fig. 6 shows the numerical results for the spatial distribution It is clear that the direction of momentum density displays swirling features along an ellipse at z = 0, and the vector field is obviously varied with the twisting phase angle field in the propagation direction. The nonplanar elliptical modes in the present method are hard to be adjusted that give rise to the inevitably high-order modes with the complex phase structures and the large angle of beam divergence shown in the first row of Fig. 2 and Fig. 3. Even though the interference pattern of nonplanar elliptical modes is difficult to be validated in the experiment, the excellent agreement of circular geometric modes between the results in Ref [37]. and Fig. 7(e) confirms the above analysis in vortex structures of nonplanar elliptical modes is reliable.

Conclusions
In summary, the diode-pumped solid-state laser under the selective pumping has been employed to excite nonplanar elliptical modes with huge fractional OAM near the degenerate cavities. The temporal dynamics for the output emission of nonplanar elliptical modes reveals to obtain the self-mode-locked states traveling along the nonplanar ray orbits inside the cavity. It has been theoretically indicated for manifesting the influence of the radial asymmetry on the vortex structures of nonplanar elliptical modes that the primary singularities can be separated by controlling the relative phase between the degenerate eigenmodes. We also numerically perform that the vector field of transverse lasing modes is displayed swirling features along an ellipse at the beam waist, and remarkably altered with the twisting phase angle field in the propagation direction.